Question

How do you find the discriminant and how many and what type of solutions does `3x^2- 2x = 5` have?

Answer 1

`y = 3x^2 - 2x - 5 = 0.`
For this type of quadratic equations, you don't need to find the Discriminant.
Shortcut: When a - b + c = 0, one real roots is (-1) and the other is `(-c/a = 5/3).`
2 real roots: -1 and 5/3.

Answer 2

For a quadratic equation in the form
`ax^2 + bx + c = 0`
the discriminant is `Delta =b^2-4ac`

Convert the given equation into the "standard form"
`3x^2-2x = 5`

`rarr 3x^2-2x-5 =0`

`Delta = (-2)^2 - 4(3)(-5)`

`Delta = 4+60 = 64 = 8^2`

Since `Delta > 0`
the given equation has `2` Real solutions

The full form for roots of the quadratic
`(-b+-sqrt(Delta))/(2a)`

would become
`(2+-8)/(2(6))`
and both solutions would be Rational (but not Integers)